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much memory. An older version of this page
using the MPEG animation format is available, but no longer actively
maintained, and so not recommended.

This animation
expands upon the classic calculus diagram above. The
diagram illustrates the local accuracy of the tangent line approximation
to a smooth curve, or--otherwise stated--the closeness of the differential
of a function to the difference of function values due to a small increment of the independent variable.
(In the diagram the increment of the independent variable
is shown in green, the differential--i.e., the product of the
derivative and the increment--in red, and the difference of function values
as the red segment plus the yellow segment. The point is
that if the green segment is small, the yellow segment is
very small.) A problem with the diagram is that when it is
drawn large enough to be visible the increment is too large to make the
point. For example, here the yellow segment is about 30% of the green
segment. This animation overcomes that problem by showing two views of
the diagram, each changing as the increment varies. In the left view
the ``camera'' is held fixed, and so the diagram becomes very small,
while in the right view the ``camera'' zooms in so that the diagram
occupies a constant area on the screen, and the relationship between the
segment lengths can be clearly seen. Note how the yellow segment
becomes very small in the second view (while the green segment appears
to be of constant length due to the zoom). Note also that as we move in
the difference between the purple curve and the blue tangent line
becomes insignificant.

These images concern the computation of a volume by integrating
cross-sectional areas. The first image reviews the basic principle. The
other images treat a specific volume, that of the wedge of water formed
when a cylindrical class of equal height and diameter is tipped until
the water line runs through the center of the base. The pictures are
frozen frames from AVS, and can only convey a rough idea of the
interactive classroom presentation (which typically lasts about 30
minutes).

In the third century B.C., Archimedes calculated the value of
to an accuracy of one accuracy of one part in a thousand. His
technique was based on inscribing and circumscribing polygons in a
circle, and is very much akin to the method of lower and upper sums
used to define the Riemann integral. His approach is presented in the
following sequence of slides.

As a way to help students appreciate functions, their applications,
and their graphs, I involve them in a small project to describe the
functions determined by the height of a bouncing ball. Although I start
by dropping a real tennis ball from one meter above ground, a better
quantitative idea of the function can be obtained from a computer
animation, including a meter stick and clock. The students view the
animation (in slow motion, with manual frame advance, etc.) and try to
construct the graph of the function. As a homework assignment they are
asked to determine the function algebraically. This is a piecewise
quadratic and helps the students to realize that piecewise defined
functions do exist outside of calculus books.

This is a pretty straightforward animation depicting the geometric
convergence of secant lines to the tangent line. The slope of the secant
(which converges to the derivative) is also displayed. I use various
variations on this demo during the early part of a calculus course.

This animation
is a version of the common demonstration that a smooth curve becomes
indistinguishable from its tangent line when viewed under a sufficiently
high power microscope. Students can easily demonstrate this themselves
using a graphics calculator equipped with a zoom button. In this
animation, we provide some extra distance queueing by showing the grid
and striping the tangent line.
Here is the Mathematica file used to
construct the animations.

An elegant geometric proof which is well within the reach of a
beginning calculus student is the proof of the fundamental trigonometric
limit

The proof is based on a diagram depicting a circular sector in the
unit circle together with an inscribed and a circumscribed triangle.
From the fact that the sector has area exceeding that of the inscribed
triangle but less than that of the circumscribed one is lead to the
inequalities

The proof then follows from the "squeeze theorem."

I usually spend about 15 minutes on this proof, including lots of
class participation. The diagram is built up in three steps: first the
sector only, then
with the inscribed triangle, and finally
with both triangles.
Here are some instructions for creating
it in class. During the presentation I make frequent recourse to plotting
software to verify the various inequalities. For example this
plot,
constructed from this MATLAB file,
convincingly verifies the second set of inequalities.

These are some simple graphs which
are useful in a discussion of limits. The first three functions all have
limit -5 as
x approaches 1, emphasizing the irrelevance of the value of the
function at the limit point itself. The last function has different
left and right hand limits at 1, and so the limit does not exist. The
graphs were constructed with this MATLAB
file.

A brief graphical exploration of a continuous, nowhere
differentiable function fits very well in the first semester of
calculus, for example, to provide a strong counterexample to the
converse of the theorem that differentiability implies continuity; or to
show that it is only differentiable functions which look like straight
lines under the microscope. Given good classroom graphics facilities
such an exploration is easy, but it is almost hopeless without them.
This plot of such a function was
produced with a few lines of Matlab code
following Weierstrass's classical construction. In class I zoom in on
this graph several times to reveal its fractal nature. Consequently I
used a very fine point spacing. On a slower machine it is preferable to
use fewer points, and decrease the point spacing as you zoom in.

Students are often puzzled by the appearance of the number e,
which is given above (to 35 decimal places). A simple explanation of
its origin arises from the fact that e is the only number for
which the tangent to the graph of y=ex
through the point (0,1) has slope exactly 1. The important result
that the function
f(x)=ex
is its own derivative follows easily from this fact and the elementary
laws of exponents. This
animation
here simply shows the graph of
y=ax,
but with varying a. By manipulating the frame advance, you can
adjust a so that the tangent has slope close to 1. The
second
animation is similar to the first, but drawn on a larger
scale, and from it one can read off the first few decimal places of e.
Here is the Mathematica file used to
construct the animations.